Question

Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................

Hint:

For triangular matrices, determinants equal the product of diagonal elements — a key simplification in such problems.

Answer 1

Correct Answer: -216

Step 1: Recognize that determinant of a power is the power of the determinant.
\[ \det((8I - M)^3) = (\det(8I - M))^3. \]

Step 2: Note that \( M \) is upper triangular.
So, \( \det(8I - M) = \prod_{i=1}^4 (8 - m_{ii}) = (8 - 9)(8 - 7)(8 - 11)(8 - 0). \)

Step 3: Simplify.
\[ \det(8I - M) = (-1)(1)(-3)(8) = (-1 \times 1 \times -3 \times 8) = 24. \] Then, \[ \det((8I - M)^3) = (24)^3 = 13824. \]

Final Answer: \[ \boxed{13824} \]